python-tuts/beginner/Calculus with Python.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Integral Exercise # 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0*x**3 + 1.0*x"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import sympy as sp\n",
    "x = sp.Symbol('x')\n",
    "sp.integrate(3.0*x**2 +1, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(10.000000000000002, 1.1102230246251568e-13)\n",
      "10.000000000000002\n"
     ]
    }
   ],
   "source": [
    "# Doing numerically\n",
    "from scipy.integrate import quad\n",
    "#lets define a function\n",
    "\n",
    "def f(y):\n",
    "    return 3.0*y**2 +1\n",
    "#print(f(2)) -> test it if it work\n",
    "\n",
    "i = quad(f,0,2)\n",
    "print(i)\n",
    "print(i[0])\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "10.000000000000002\n",
      "1.1102230246251568e-13\n"
     ]
    }
   ],
   "source": [
    "i, error = quad(f, 0, 2)\n",
    "print(i)\n",
    "print(error)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$f(x) = \\int_{0}^{2\\pi}e^{-x}{sin(3x)} dx$$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# try yourself and plot this with you ipynb\n",
    "# play around with http://wolframalpha.com, cool site for mathematics\n",
    "from IPython.display import display, Math, Latex\n",
    "display(Math(r'f(x) = \\int_{0}^{2\\pi}e^{-x}{sin(3x)} dx'))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(0.29943976718048754, 5.05015300411582e-13)\n"
     ]
    }
   ],
   "source": [
    "#Solution\n",
    "from scipy.integrate import quad\n",
    "import numpy as np\n",
    "# import sympy as sp - this wont really work, you can and should try throwing in numbers to see why its deosn't!\n",
    "\n",
    "def f(x):\n",
    "    return np.exp(-x)*np.sin(3.0*x)\n",
    "i = quad(f, 0, 2*np.pi)\n",
    "print(i)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 2:  Simple derivatives or as I'd prefer to call differentials "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.0009999999996975\n",
      "4.000999999999699\n",
      "6.000999999999479\n",
      "8.0010000000037\n"
     ]
    }
   ],
   "source": [
    "# Finding derivative the simple math way\n",
    "\n",
    "def f(x):\n",
    "    return x**2\n",
    "\n",
    "def differential(x):\n",
    "    h =  1.0/1000.0\n",
    "    rise = f(x + h) - f(x)\n",
    "    run = h\n",
    "    slope = rise / run\n",
    "    return slope\n",
    "\n",
    "print(differential(1))\n",
    "print(differential(2))\n",
    "print(differential(3))\n",
    "print(differential(4))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Area is:  0.2850000000000001\n",
      "Area is:  0.33283350000000034\n",
      "Area is:  0.33332833334999745\n"
     ]
    }
   ],
   "source": [
    "# Integrals are essentially the area under the sinusoidal curve tha covers \n",
    "# all reactangular boxes undr the curve covered\n",
    "def integral(startx, endx, numberofrectangles):\n",
    "#     width = ((endx) - (startx)) /numberofrectangles = Makes no difference in Py 3.6, with or without float\n",
    "    width = (float(endx) - float(startx)) /numberofrectangles\n",
    "    runningSum = 0\n",
    "    for i in range(numberofrectangles):\n",
    "        height = f(startx + i*width)\n",
    "        area = height*width\n",
    "        runningSum += area\n",
    "    return runningSum\n",
    "print('Area is: ', integral(0, 1, 10))\n",
    "print('Area is: ', integral(0, 1, 1000))\n",
    "print('Area is: ', integral(0, 1, 100000))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 3: More integration, interpolation and curve fitting"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "from scipy.integrate import *"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### You've seen one example of integra before, but lets try another one "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Try solving this: \n",
    "$f(x) = \\int_{0}^{1}3x^2 + {1} dx$\n",
    "#### Also BTW just use dollar sign before and after your formula as this $your formula$\n",
    "to display formulas in your iPython Notebook use the inline function such as \n",
    "$$e^x=\\sum_{i=0}^\\infty \\frac{1}{i!}x^i$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Integral is:  2.0\n",
      "Error is : 2.220446049250313e-14\n"
     ]
    }
   ],
   "source": [
    "def f(x):\n",
    "    return 3.0*x**2 + 1.0\n",
    "xlow = 0\n",
    "xhigh = 1\n",
    "i, err = quad(f, xlow, xhigh)\n",
    "print(\"Integral is: \", i)\n",
    "print(\"Error is :\", err)\n",
    "# Use help(quad) to see what more it can do"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Let's try interpolation "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x111249be0>"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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Muw3Se6mt5szcuRUMZ17EHZBEpyhbK+IN2xn0YzCgNPXkIH+AYSyFFwcqK/st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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11075c630>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.interpolate import * # We want interpid function\n",
    "plt.xkcd()\n",
    "\n",
    "# Here basically you have a series of data points.\n",
    "# You want to estimate data points that are between given points\n",
    "# you want to *linearly* interpolate between the given function (meaning drawing a straight line)\n",
    "# Or, you can use higher order, called polynomial interpolation to approximate the curve between the data points\n",
    "# Cubic spline is a popular example of this\n",
    "\n",
    "x_givenpoint = np.linspace(0, 10, 10)\n",
    "y_givenpoint = np.cos(x_givenpoint**2.0/8.0)\n",
    "\n",
    "\n",
    "\n",
    "# Now we create an array\n",
    "xx = np.linspace(0, 10, 1000)\n",
    "yy = np.cos(xx**2.0/8.0)\n",
    "\n",
    "plt.plot(x_givenpoint, y_givenpoint, 'x', label = 'given data')\n",
    "plt.plot(xx, yy, ':', label='Perfect')\n",
    "plt.ylabel('x')\n",
    "plt.xlabel('y')\n",
    "plt.legend(loc='best')\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x10fae8da0>"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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Oe9Qd7cfADV/BdV8iuOCODncAsDx7OdKEVtjOnMF86BD24uIACxWpC5zV1VjP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\ncDhTwoMEh8PhcKaEBwkOh8PhTMn/AGg8KNnYyFhBAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11074a400>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x_i = np.linspace(0, 10, 100) #\n",
    "\n",
    "# do linear interpolation\n",
    "f_linear = interp1d(x_givenpoint, y_givenpoint)\n",
    "y_il = f_linear(x_i)\n",
    "\n",
    "# Cubic spline interpolation\n",
    "\n",
    "f_spline = interp1d(x_givenpoint, y_givenpoint, kind='cubic')\n",
    "y_is = f_linear(x_i)\n",
    "\n",
    "# plot these results\n",
    "plt.plot(x_givenpoint, y_givenpoint, 'x')\n",
    "plt.plot(x_i, y_il, 'o')\n",
    "plt.plot(x_i, y_is, '-')\n",
    "plt.plot(xx, yy, ':')\n",
    "plt.legend(['data', 'linear', 'perfect'], loc='best')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Curve fitting - how does it look like?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-2, 2)"
      ]
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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P7j5w8RuamNp+HPQMx+KYW1kMFY4q9K/kGwLnvwgXvAJ5g6H8G3hiCnz0IIjf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\nQi/h/wPXZsO5G2kdewAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11129cb70>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x_given = np.array([0., 1., 2., 3., 4., 5.])\n",
    "y_given = np.array([0, 0.8, 0.9, 0.1, -0.8, -1.0])\n",
    "\n",
    "x_p = np.linspace(-2.0, 6.0, 100)    # Data for plotting the pply fit\n",
    "p3 = np.polyfit(x_given, y_given, 3) # 3rd order polynomial object\n",
    "y_p = np.polyval(p3, x_p)\n",
    "plt.plot(x_given, y_given, 'o')\n",
    "plt.plot(x_p, y_p, '-')\n",
    "plt.legend(['data', 'polyfit'], loc='best')\n",
    "plt.ylim(-2, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.08703704 -0.81349206  1.69312169 -0.03968254]\n"
     ]
    }
   ],
   "source": [
    "print(p3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Lets do some curve fitting\n",
    "This function is available from scipy.optimizer import * \n",
    "Fit function from scipy.org \n",
    "$f(x) = a^-bx + c$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from scipy.optimize import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "a=2.29404, b=1.49633, c=0.802552\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x1140c2518>"
      ]
     },
     "execution_count": 105,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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VMpoogEoFnYb6P6+bVj3nEAgiBI1GUy6ypeDcQJKkcvMUzrnRRMBZJ3OuFO3v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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11167af98>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def f(x, a, b, c):\n",
    "    return a*np.exp(-b*x) + c\n",
    "\n",
    "# Setting some given data\n",
    "\n",
    "x_given = np.linspace(0,4,75)\n",
    "y_given = f(x_given,2.5,1.5,0.8) + 0.2*np.random.normal(size=len(x_given))\n",
    "\n",
    "# do some curve fitting\n",
    "\n",
    "params, extras = curve_fit(f, x_given, y_given)\n",
    "\n",
    "# output / plotting results\n",
    "\n",
    "print(\"a=%g, b=%g, c=%g\" %(params[0], params[1], params[2]))\n",
    "\n",
    "plt.plot(x_given, y_given, 'o')\n",
    "plt.plot(x_given, f(x_given, params[0], params[1], params[2]))\n",
    "plt.legend(['data', 'curve-fit'], loc='best')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Some more simple calculus stuff\n",
    "Go to https://sympy.org for more basics"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing() # prints cool output"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEoAAAAVCAYAAADhCHhTAAAABHNCSVQICAgIfAhkiAAAA2lJREFU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\nF0v77Vppf5vCFP4j/A1ZjxS9AKMJKgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$x \\cos{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "x⋅cos(x)"
      ]
     },
     "execution_count": 114,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x, y, z = symbols('x,y,z')\n",
    "my_ex = x*cos(x)\n",
    "my_ex"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAD8AAAAVCAYAAADxaDaPAAAABHNCSVQICAgIfAhkiAAAAzZJREFU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\nP0nbxbV7tJglethS9RPoNP5X+Bvixs3MIL1ItAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\cos{\\left (1 \\right )}$$"
      ],
      "text/plain": [
       "cos(1)"
      ]
     },
     "execution_count": 115,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# substituting this function\n",
    "my_ex.subs(x, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEoAAAAVCAYAAADhCHhTAAAABHNCSVQICAgIfAhkiAAAAwpJREFU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\n",
      "text/latex": [
       "$$- \\sin{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "-sin(x)"
      ]
     },
     "execution_count": 117,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(cos(x), x) #diff is the derivative function in sympy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADYAAAAbCAYAAAA3d3w1AAAABHNCSVQICAgIfAhkiAAAArBJREFU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\n7kquFxJqXDsAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$2 x e^{x^{2}}$$"
      ],
      "text/plain": [
       "     ⎛ 2⎞\n",
       "     ⎝x ⎠\n",
       "2⋅x⋅ℯ    "
      ]
     },
     "execution_count": 118,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(exp(x**2), x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACgAAAASCAYAAAApH5ymAAAABHNCSVQICAgIfAhkiAAAAeRJREFU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\nSC1ONoMLpBfpPZbkcNYH50GB+P/xb+EXb3uP8bSCw/UAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$24 x$$"
      ],
      "text/plain": [
       "24⋅x"
      ]
     },
     "execution_count": 120,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(x**4, x, x, x) #can take multiple derivatives at once\n",
    "diff(x**4, x, 3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Below will compute the following\n",
    "\n",
    "$\\frac{\\partial 7}{\\partial x \\partial y^2 \\partial z^4}e^{xyz}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 122,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAZ8AAAAaCAYAAABo6FFLAAAABHNCSVQICAgIfAhkiAAACr9JREFU\neJztnXuwVVUdxz+XR4iPULCEnMIp0ihJqmEYjZePcIicKCmTqTgMjdWkRoSC5OOMM3apTCBJodK5\nltGoJL3MQMwQ6WGMEiYPH3QtE8SbpBQIGLc/vmtz991377PX2Xuvc8+5rM/MHjh7rfXbv7V+a++1\n1m/99r7g8Xg8Hk+D8SVgE/CqOf4ATC5A7lXAn43Ml4BfAqdblGvKeV1X9XEh15WuPYWsfag75LrS\n1XNkkPe515B8FJgEDANOBW4ADgLvzSl3FTAD3YAjgJXATmBghTLnAm/PeV1X9XEh15WuPYUsfai7\n5LrS1XNk8DHgxO5Woh54Gfh8wTKPBf4HXJCQ/k7guoKvGeCiPq7kutK1J5DWh+pJritdPT2TNwBL\ngT4FypwK7AeGhs4tBp4FTgKeB2ZHyowAXgPeDVwJtMcc1xeo42F6A58CDhglimQIUnxMQvoK4I0F\nX9NVfVzIddn2PYW0PlRPcl3p6um5TAG+WKC8JmAD8H3zew6wC030Ae4BfhIp8yCwxPz/OGBw6LgR\n2IE8NYUxAvgP8Drwb9zsO9wNPI4eslE+CPyowGu5qo8LubVoexumAjcD69C+RTtwZ5UyPk3H7Ohz\nhWonKvWhepPrSlcbWomfsbYjV2CUQcheK4FngH3AK8AjwEygl3ONGwvbfj4ZWI1WGPuA7eiBf2ZC\n/r7ARqB/YZrCROTKn4fu61GhtNnI3gFTkOdlUIycucA/gdMK1A3Qkm8Y8AGgGWij2M3Sm4AXSN7P\nuY9iH7qu6uNCruu2t2Ujupn2AFuofvB5Kxo89+Bm8EnrQ/Uk15WutrQiW5Rjjjkx+b+AbPYC8GPU\nD283MtqRV+KI3BCPwbaff8OktwE/ABagdjwAHEIDWBy3oECkIvk9mtxOipw/y+g4EOiHBqJZMeWv\nQgPoqQXrFcsa4LaCZC1ES7V3JaQPQiOzy822IuvjWm5WmSXUkSZkvO7ZaDneZGRUM/g0Ib2fBb5F\n8YNPWh+qJ7lFyCyRz5at5rDlHLQ3FV3hDAb+bnS5MKMu9UKJfG0K9v18MNrv2wm8OZJ2tim3PeEa\n04H1OXSMcg7wX6PP+yNp/dCe0ES0stmKVl9hrkYDT6yrLdphVhPfWZqAFpO2IEXhXkaxgNmm3FcT\n8p+GKvFw5Pxi4GLUAFsTyo5Hfsi2hPR6qk+a3IBHSHZ7tKfITZLpmoeAp5F+1XI5svEM1NGTyGJL\nmz6UxZ42cqE6W9rKrDd+i0LDD0XO70Sb4BD/0La1Zxb75LmHXGHbz4ei+/hP6NkW5iG0anpTQtmN\nwGgUsBLHhcD96Hl5AN2z84l3756BXKmXAT9DK9ow+5Fr+ALga2hVfDCUfi1wCXpGP4MFZ6BRbnNE\noW8joy2L5F8AjAVOQfsPzagThpdoo03ZexKuuRot68Ihwt9FPsZz6LxpFW3UG5BB6r0+NnID5hDv\n8njOXLecQWYaJfLP7AImYL/yGY782QvN7zLJM8JqbWnbh6q1p61csLdlNTLTKJF/5bMDuXbmA19G\nM+4s+09XGF0WxqTZ2jPL/Wbb7raUyNem1fTzgejBvoOuHp1xptzKhOscRXygSm8UHNCOBpxbgUXA\nNnPujkj+ociNeq35fTp6tkyI5Ftozq+KnL8aDXBn0bk/H5Wg92FajEIl83u++X0XXVdKLcig+9Eo\nvQY4P5KnL7AXLcGjfMLIXhw5nzRjKUfyLUejcr3Xx0ZuJW40cm8P6ZxXZpgStR98+qBomm10bJKW\nqex2a8HelrZ9qFp72spNIs6WeWWGKZF/8InTZTuaxdrSB3jClE3qly2k2zPL/RZHXLvbUiJ7m2bp\n57PQQ30X8D00sbwbhTGvpqs7Lsx+4DORc0vM9ZrpHI7dF7np2lF4NGjw20LXydxd6EX2MNPRwP+e\n0LkmFHAS14fOraA3oE2xfcDfgEtNod+gze2srDVyhoTOHQP8A3gRGJBR7v2kR7o1Un2iNKFNxHbU\ngVxt3Jao/eBzPZr5hqN3ylS+KV3YEmpjz0ax5XVoBXYScDSa9S5FD8O9aMViQ/Cwv69CHlt75rFP\nEe1eInubZunn0BE9Fn54Pw1MS7leG3LxBYxGtkuapF9iZM9IkRvHA2jVXijNdFR4PeqEefi6kfXx\n0LkgoqOUQ+6DdMShV6JR6hOmN1oOtxvZRdFKZX949GipUv4E0gef0WjG9M3I+TLpN2XRtgT39mxU\nW4YJBpMkl0+Yy03eLaR/pcHGnlntk6XdWymuTbP28ytNuZtQ1OPRaMN/lSkXlRfmeTpHJd5pyiwn\n3hW5wqTPrCAzTC80MZmLXIMnWJbrQtIbsS+F/j8TzXjyEERgjAbuRdE8X0HLuKi/sRr2YucXb5T6\nBPRFnWUqmokW+VbwIuD4yLmR6HM9d9A10mljgdcG9bkfAk8B12QoX7Qtwa09e4otl6JN/3Ep+S5F\nbrDNyM3yckp+G3tmsU/Wdi+qTbP28wlooFxJ5y8IPIY+o/MUssNS4qPejqNzG040/16cct3nLPUb\nh4JMtqEAht2W5ayYhpZpO9CIeGsBMk8wMtea32vQUvR9OeXeRuVlPTRWfUCbcr9CuiZF+BRNidq5\n3Y7Hfla5KFLWhS3BnT0b3ZZhBhi5r1XIM8vkeYLK+xIBtvas1j5Ft3uJ6ts0az8PVpiXJci916TH\nha83oXYJVohBAMLamLx1x4dRCN4mFM63FYXPFfFm6pPobfxpqEFuKUDmPOCPFdIbrT7HIFfiIYr9\nVEYaJWo3+PRHL87FHY+ZsuvM74tC5VzaEoq3Z0+wZZjzjdzNCelzTfrj2L13V609be3jot1LVN+m\nWfv5zSYtaaW2zqTHffdvoEkLXjTvb34/WYXe3cIYtFzbTsfG3lSkfFpEmQ3LjKw9aKmd2VcYYoKR\nF7eJ2Gj1GYDcC68Dn80pq1pK1G7wqUSZeF+4a1tCsfZsVFsORw/vKKfQ8R7X/Jj0a0zaBuy+xJ3F\nnjb2cdXuJYod0Msk7/l80qTtBE6OpE1Cg+o+4j9jMx59QSEczfcXuu6XhRlD93zG6fCez0i0TH0F\n+BBaBoM2ozYgf+dYNOpmZT2KrDgW+WuL8BUGn354B51fZGrE+ixHcfGPok3GckyeZhRKWW9MMQco\nnh8U3dNi/t9G/KdZbKiFLaFYezaqLS9CbqqH0R7AHnRvTUYunF8jt1CY6XREdK2jc6RVQCsdfSGr\nPW3s06jtHmYFciuehwI2gj+vMRz4CJpozwP+FVN2JIoADr/wewXamvipkbsJDU4no89y9QXe5qAe\nVgxDldtN/N+COQ+NnJXcWzaMNXIepdgw0++gThnQiPXpRcf3npKOF3NeoxIl8s3sylTWvbVKOcGM\nsFa2hOLs2ci2HI9eRtyKZtAH0SrjAbSSiGuXMul7Gr8zefPYM80+Ltu9RO1WPqABYRZqh1fRBHsX\nGrQnJpQB+Dlqwyij0KC2E9m0De3LLcPi3ZuewC/Q7GhUWsYqGULXN2xrgav6eLoHb8/6xtunMiei\nlY0nQrBJuCQtY0aaqe3XnF3Xx1NbvD3rG2+fdOai1aEH+RHnoZdA9wN/pZgXAuPoh2LjXX66vZb1\n8bjH27O+8faxZyi1C+VvCILPNuxG3yV6i+PrDcbt30CpdX08bvH2rG+8few5E/83kzwej8fj8Xg8\nHo/H4/F4PB4P/B8QKNdz6vXofgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$x^{3} y^{2} \\left(x^{3} y^{3} z^{3} + 14 x^{2} y^{2} z^{2} + 52 x y z + 48\\right) e^{x y z}$$"
      ],
      "text/plain": [
       " 3  2 ⎛ 3  3  3       2  2  2                ⎞  x⋅y⋅z\n",
       "x ⋅y ⋅⎝x ⋅y ⋅z  + 14⋅x ⋅y ⋅z  + 52⋅x⋅y⋅z + 48⎠⋅ℯ     "
      ]
     },
     "execution_count": 122,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = exp(x*y*z)\n",
    "diff(expr, x,y,y,z,z,z,z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 123,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAZ8AAAAaCAYAAABo6FFLAAAABHNCSVQICAgIfAhkiAAACr9JREFU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\nHo/H4/F4PB4P/B8QKNdz6vXofgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$x^{3} y^{2} \\left(x^{3} y^{3} z^{3} + 14 x^{2} y^{2} z^{2} + 52 x y z + 48\\right) e^{x y z}$$"
      ],
      "text/plain": [
       " 3  2 ⎛ 3  3  3       2  2  2                ⎞  x⋅y⋅z\n",
       "x ⋅y ⋅⎝x ⋅y ⋅z  + 14⋅x ⋅y ⋅z  + 52⋅x⋅y⋅z + 48⎠⋅ℯ     "
      ]
     },
     "execution_count": 123,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(expr, x,y,y,z,4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 124,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAZ8AAAAaCAYAAABo6FFLAAAABHNCSVQICAgIfAhkiAAACr9JREFU\neJztnXuwVVUdxz+XR4iPULCEnMIp0ihJqmEYjZePcIicKCmTqTgMjdWkRoSC5OOMM3apTCBJodK5\nltGoJL3MQMwQ6WGMEiYPH3QtE8SbpBQIGLc/vmtz991377PX2Xuvc8+5rM/MHjh7rfXbv7V+a++1\n1m/99r7g8Xg8Hk+D8SVgE/CqOf4ATC5A7lXAn43Ml4BfAqdblGvKeV1X9XEh15WuPYWsfag75LrS\n1XNkkPe515B8FJgEDANOBW4ADgLvzSl3FTAD3YAjgJXATmBghTLnAm/PeV1X9XEh15WuPYUsfai7\n5LrS1XNk8DHgxO5Woh54Gfh8wTKPBf4HXJCQ/k7guoKvGeCiPq7kutK1J5DWh+pJritdPT2TNwBL\ngT4FypwK7AeGhs4tBp4FTgKeB2ZHyowAXgPeDVwJtMcc1xeo42F6A58CDhglimQIUnxMQvoK4I0F\nX9NVfVzIddn2PYW0PlRPcl3p6um5TAG+WKC8JmAD8H3zew6wC030Ae4BfhIp8yCwxPz/OGBw6LgR\n2IE8NYUxAvgP8Drwb9zsO9wNPI4eslE+CPyowGu5qo8LubVoexumAjcD69C+RTtwZ5UyPk3H7Ohz\nhWonKvWhepPrSlcbWomfsbYjV2CUQcheK4FngH3AK8AjwEygl3ONGwvbfj4ZWI1WGPuA7eiBf2ZC\n/r7ARqB/YZrCROTKn4fu61GhtNnI3gFTkOdlUIycucA/gdMK1A3Qkm8Y8AGgGWij2M3Sm4AXSN7P\nuY9iH7qu6uNCruu2t2Ujupn2AFuofvB5Kxo89+Bm8EnrQ/Uk15WutrQiW5Rjjjkx+b+AbPYC8GPU\nD283MtqRV+KI3BCPwbaff8OktwE/ABagdjwAHEIDWBy3oECkIvk9mtxOipw/y+g4EOiHBqJZMeWv\nQgPoqQXrFcsa4LaCZC1ES7V3JaQPQiOzy822IuvjWm5WmSXUkSZkvO7ZaDneZGRUM/g0Ib2fBb5F\n8YNPWh+qJ7lFyCyRz5at5rDlHLQ3FV3hDAb+bnS5MKMu9UKJfG0K9v18MNrv2wm8OZJ2tim3PeEa\n04H1OXSMcg7wX6PP+yNp/dCe0ES0stmKVl9hrkYDT6yrLdphVhPfWZqAFpO2IEXhXkaxgNmm3FcT\n8p+GKvFw5Pxi4GLUAFsTyo5Hfsi2hPR6qk+a3IBHSHZ7tKfITZLpmoeAp5F+1XI5svEM1NGTyGJL\nmz6UxZ42cqE6W9rKrDd+i0LDD0XO70Sb4BD/0La1Zxb75LmHXGHbz4ei+/hP6NkW5iG0anpTQtmN\nwGgUsBLHhcD96Hl5AN2z84l3756BXKmXAT9DK9ow+5Fr+ALga2hVfDCUfi1wCXpGP4MFZ6BRbnNE\noW8joy2L5F8AjAVOQfsPzagThpdoo03ZexKuuRot68Ihwt9FPsZz6LxpFW3UG5BB6r0+NnID5hDv\n8njOXLecQWYaJfLP7AImYL/yGY782QvN7zLJM8JqbWnbh6q1p61csLdlNTLTKJF/5bMDuXbmA19G\nM+4s+09XGF0WxqTZ2jPL/Wbb7raUyNem1fTzgejBvoOuHp1xptzKhOscRXygSm8UHNCOBpxbgUXA\nNnPujkj+ociNeq35fTp6tkyI5Ftozq+KnL8aDXBn0bk/H5Wg92FajEIl83u++X0XXVdKLcig+9Eo\nvQY4P5KnL7AXLcGjfMLIXhw5nzRjKUfyLUejcr3Xx0ZuJW40cm8P6ZxXZpgStR98+qBomm10bJKW\nqex2a8HelrZ9qFp72spNIs6WeWWGKZF/8InTZTuaxdrSB3jClE3qly2k2zPL/RZHXLvbUiJ7m2bp\n57PQQ30X8D00sbwbhTGvpqs7Lsx+4DORc0vM9ZrpHI7dF7np2lF4NGjw20LXydxd6EX2MNPRwP+e\n0LkmFHAS14fOraA3oE2xfcDfgEtNod+gze2srDVyhoTOHQP8A3gRGJBR7v2kR7o1Un2iNKFNxHbU\ngVxt3Jao/eBzPZr5hqN3ylS+KV3YEmpjz0ax5XVoBXYScDSa9S5FD8O9aMViQ/Cwv69CHlt75rFP\nEe1eInubZunn0BE9Fn54Pw1MS7leG3LxBYxGtkuapF9iZM9IkRvHA2jVXijNdFR4PeqEefi6kfXx\n0LkgoqOUQ+6DdMShV6JR6hOmN1oOtxvZRdFKZX949GipUv4E0gef0WjG9M3I+TLpN2XRtgT39mxU\nW4YJBpMkl0+Yy03eLaR/pcHGnlntk6XdWymuTbP28ytNuZtQ1OPRaMN/lSkXlRfmeTpHJd5pyiwn\n3hW5wqTPrCAzTC80MZmLXIMnWJbrQtIbsS+F/j8TzXjyEERgjAbuRdE8X0HLuKi/sRr2YucXb5T6\nBPRFnWUqmokW+VbwIuD4yLmR6HM9d9A10mljgdcG9bkfAk8B12QoX7Qtwa09e4otl6JN/3Ep+S5F\nbrDNyM3yckp+G3tmsU/Wdi+qTbP28wlooFxJ5y8IPIY+o/MUssNS4qPejqNzG040/16cct3nLPUb\nh4JMtqEAht2W5ayYhpZpO9CIeGsBMk8wMtea32vQUvR9OeXeRuVlPTRWfUCbcr9CuiZF+BRNidq5\n3Y7Hfla5KFLWhS3BnT0b3ZZhBhi5r1XIM8vkeYLK+xIBtvas1j5Ft3uJ6ts0az8PVpiXJci916TH\nha83oXYJVohBAMLamLx1x4dRCN4mFM63FYXPFfFm6pPobfxpqEFuKUDmPOCPFdIbrT7HIFfiIYr9\nVEYaJWo3+PRHL87FHY+ZsuvM74tC5VzaEoq3Z0+wZZjzjdzNCelzTfrj2L13V609be3jot1LVN+m\nWfv5zSYtaaW2zqTHffdvoEkLXjTvb34/WYXe3cIYtFzbTsfG3lSkfFpEmQ3LjKw9aKmd2VcYYoKR\nF7eJ2Gj1GYDcC68Dn80pq1pK1G7wqUSZeF+4a1tCsfZsVFsORw/vKKfQ8R7X/Jj0a0zaBuy+xJ3F\nnjb2cdXuJYod0Msk7/l80qTtBE6OpE1Cg+o+4j9jMx59QSEczfcXuu6XhRlD93zG6fCez0i0TH0F\n+BBaBoM2ozYgf+dYNOpmZT2KrDgW+WuL8BUGn354B51fZGrE+ixHcfGPok3GckyeZhRKWW9MMQco\nnh8U3dNi/t9G/KdZbKiFLaFYezaqLS9CbqqH0R7AHnRvTUYunF8jt1CY6XREdK2jc6RVQCsdfSGr\nPW3s06jtHmYFciuehwI2gj+vMRz4CJpozwP+FVN2JIoADr/wewXamvipkbsJDU4no89y9QXe5qAe\nVgxDldtN/N+COQ+NnJXcWzaMNXIepdgw0++gThnQiPXpRcf3npKOF3NeoxIl8s3sylTWvbVKOcGM\nsFa2hOLs2ci2HI9eRtyKZtAH0SrjAbSSiGuXMul7Gr8zefPYM80+Ltu9RO1WPqABYRZqh1fRBHsX\nGrQnJpQB+Dlqwyij0KC2E9m0De3LLcPi3ZuewC/Q7GhUWsYqGULXN2xrgav6eLoHb8/6xtunMiei\nlY0nQrBJuCQtY0aaqe3XnF3Xx1NbvD3rG2+fdOai1aEH+RHnoZdA9wN/pZgXAuPoh2LjXX66vZb1\n8bjH27O+8faxZyi1C+VvCILPNuxG3yV6i+PrDcbt30CpdX08bvH2rG+8few5E/83kzwej8fj8Xg8\nHo/H4/F4PB4P/B8QKNdz6vXofgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$x^{3} y^{2} \\left(x^{3} y^{3} z^{3} + 14 x^{2} y^{2} z^{2} + 52 x y z + 48\\right) e^{x y z}$$"
      ],
      "text/plain": [
       " 3  2 ⎛ 3  3  3       2  2  2                ⎞  x⋅y⋅z\n",
       "x ⋅y ⋅⎝x ⋅y ⋅z  + 14⋅x ⋅y ⋅z  + 52⋅x⋅y⋅z + 48⎠⋅ℯ     "
      ]
     },
     "execution_count": 124,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(expr, x,y,2,z,4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 125,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAZ8AAAAaCAYAAABo6FFLAAAABHNCSVQICAgIfAhkiAAACr9JREFU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\nHo/H4/F4PB4P/B8QKNdz6vXofgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$x^{3} y^{2} \\left(x^{3} y^{3} z^{3} + 14 x^{2} y^{2} z^{2} + 52 x y z + 48\\right) e^{x y z}$$"
      ],
      "text/plain": [
       " 3  2 ⎛ 3  3  3       2  2  2                ⎞  x⋅y⋅z\n",
       "x ⋅y ⋅⎝x ⋅y ⋅z  + 14⋅x ⋅y ⋅z  + 52⋅x⋅y⋅z + 48⎠⋅ℯ     "
      ]
     },
     "execution_count": 125,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# It can be called as a method\n",
    "expr.diff(x,y,y,z,4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 128,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGcAAAApCAYAAAAoEzflAAAABHNCSVQICAgIfAhkiAAAA/9JREFU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\nfgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{\\partial^{7}}{\\partial x\\partial y^{2}\\partial z^{4}}  e^{x y z}$$"
      ],
      "text/plain": [
       "     7            \n",
       "    ∂     ⎛ x⋅y⋅z⎞\n",
       "──────────⎝ℯ     ⎠\n",
       "  4   2           \n",
       "∂z  ∂y  ∂x        "
      ]
     },
     "execution_count": 128,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Create unevaluated rerivative by using Derivative class\n",
    "deriv = Derivative(expr, x,y,y,z,4)\n",
    "deriv"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 129,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sympy.core.assumptions.ManagedProperties"
      ]
     },
     "execution_count": 129,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(Derivative)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 130,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAZ8AAAAaCAYAAABo6FFLAAAABHNCSVQICAgIfAhkiAAACr9JREFU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\nHo/H4/F4PB4P/B8QKNdz6vXofgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$x^{3} y^{2} \\left(x^{3} y^{3} z^{3} + 14 x^{2} y^{2} z^{2} + 52 x y z + 48\\right) e^{x y z}$$"
      ],
      "text/plain": [
       " 3  2 ⎛ 3  3  3       2  2  2                ⎞  x⋅y⋅z\n",
       "x ⋅y ⋅⎝x ⋅y ⋅z  + 14⋅x ⋅y ⋅z  + 52⋅x⋅y⋅z + 48⎠⋅ℯ     "
      ]
     },
     "execution_count": 130,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Evaluate an unevaluated derivaive using doit method\n",
    "deriv.doit()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## A bit more basics of Integrals "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$f(x) = \\int_{0}^{∞}e^{-x} dx$\n",
    "\n",
    "##### alt+5 = infinity symbol\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 131,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA0AAAASCAYAAACAa1QyAAAABHNCSVQICAgIfAhkiAAAAHZJREFU\nKJFjYKACCGFgYJjMwMBwmIGB4RMDA8N/BgaGJYQ0XYAq/MzAwHCdWE2ODAwMqgwMDIwMDAwOuDSx\noPH3EzKVgYGBgYkYRaOaBlwTeuQGQDEDAwODBJS2ZGBgWABlv2FgYChBN6SBAZJ0cOEH5LiMzgAA\n6XoX52TB9a4AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1$$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 131,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(exp(-x), (x,0, oo))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Integrate this\n",
    "$\\int_{0}^{∞}\\int_{-∞}^{∞}e^{-x^2-y^2} dxdy$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 137,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA8AAAAPCAYAAAA71pVKAAAABHNCSVQICAgIfAhkiAAAAL5JREFU\nKJHV0D1qgkEUheFHEYRAIFgHrISAFuo2bOxcQCp7cQXuwU1Y2Nu6AH9IY5UFpBEhTchPM8J49RML\nmxwYLrxz7j13hjuph98bTwcqWfMeUwzxhll294IBllhgfSm9nyZPAh8n/prDcjC1U10F3i3gJ5qn\nhEbgO3yheq35HQeUMvaIH2yjOV/7CXVsUvpR7TTsbOVyMHH+k4XvvdQcTa1Ur65dlFxLtYnnOOCo\nFb7xEPgAH/jEqKj5H+kPZCwqsiZWNEcAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\pi$$"
      ],
      "text/plain": [
       "π"
      ]
     },
     "execution_count": 137,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(exp(-x**2 - y**2), (x, -oo, oo), (y, -oo, oo))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 141,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Integral(x**x, x)\n"
     ]
    }
   ],
   "source": [
    "# if integrate isnt able to compute an integral, it returns an unevaluated Integral object\n",
    "expr = integrate(x**x, x)\n",
    "print(expr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 140,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEMAAAAgCAYAAABJqsWHAAAABHNCSVQICAgIfAhkiAAAAw5JREFU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\n",
      "text/latex": [
       "$$\\int x^{x}\\, dx$$"
      ],
      "text/plain": [
       "⌠      \n",
       "⎮  x   \n",
       "⎮ x  dx\n",
       "⌡      "
      ]
     },
     "execution_count": 140,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 143,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHEAAAAgCAYAAAAlrJeCAAAABHNCSVQICAgIfAhkiAAABXVJREFU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\n",
      "text/latex": [
       "$$\\int \\log^{2}{\\left (x \\right )}\\, dx$$"
      ],
      "text/plain": [
       "⌠           \n",
       "⎮    2      \n",
       "⎮ log (x) dx\n",
       "⌡           "
      ]
     },
     "execution_count": 143,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = Integral(log(x)**2, x)\n",
    "expr"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Itegrate uses powerful algorithms including heuristic pattern matching, Risch algorithm and Meijer G-functions\n",
    "integrate uses powerful algorithms that are always improving to compute both definite and indefinite integrals. Here's a cool example."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 145,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAR8AAAApCAYAAAAMAy1JAAAABHNCSVQICAgIfAhkiAAACKdJREFU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\nSUVORK5CYII=\n",
      "text/latex": [
       "$$\\int \\frac{\\left(x^{4} + x^{2} e^{x} - x^{2} - 2 x e^{x} - 2 x - e^{x}\\right) e^{x}}{\\left(x - 1\\right)^{2} \\left(x + 1\\right)^{2} \\left(e^{x} + 1\\right)}\\, dx$$"
      ],
      "text/plain": [
       "⌠                                            \n",
       "⎮ ⎛ 4    2  x    2        x          x⎞  x   \n",
       "⎮ ⎝x  + x ⋅ℯ  - x  - 2⋅x⋅ℯ  - 2⋅x - ℯ ⎠⋅ℯ    \n",
       "⎮ ──────────────────────────────────────── dx\n",
       "⎮               2        2 ⎛ x    ⎞          \n",
       "⎮        (x - 1) ⋅(x + 1) ⋅⎝ℯ  + 1⎠          \n",
       "⌡                                            "
      ]
     },
     "execution_count": 145,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integ = Integral((x**4 + x**2*exp(x) - x**2 - 2*x*exp(x) - 2*x - \\\n",
    "                  exp(x))*exp(x)/((x - 1)**2*(x + 1)**2*(exp(x) + 1)), x)\n",
    "integ"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 146,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAK4AAAAjCAYAAAANIjHoAAAABHNCSVQICAgIfAhkiAAABk1JREFU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\nioqKioqKioqKijz+BydTypur497WAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\log{\\left (e^{x} + 1 \\right )} + \\frac{e^{x}}{x^{2} - 1}$$"
      ],
      "text/plain": [
       "                 x  \n",
       "   ⎛ x    ⎞     ℯ   \n",
       "log⎝ℯ  + 1⎠ + ──────\n",
       "               2    \n",
       "              x  - 1"
      ]
     },
     "execution_count": 146,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integ.doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 148,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHAAAAAgCAYAAADKbvy8AAAABHNCSVQICAgIfAhkiAAABRdJREFU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\nSUVORK5CYII=\n",
      "text/latex": [
       "$$\\int \\sin{\\left (x^{2} \\right )}\\, dx$$"
      ],
      "text/plain": [
       "⌠           \n",
       "⎮    ⎛ 2⎞   \n",
       "⎮ sin⎝x ⎠ dx\n",
       "⌡           "
      ]
     },
     "execution_count": 148,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integ = Integral(sin(x**2), x)\n",
    "integ"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 149,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALwAAAAoCAYAAABEvpD+AAAABHNCSVQICAgIfAhkiAAACblJREFU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\n",
      "text/latex": [
       "$$\\frac{3 \\sqrt{2} \\sqrt{\\pi} S\\left(\\frac{\\sqrt{2} x}{\\sqrt{\\pi}}\\right)}{8 \\Gamma{\\left(\\frac{7}{4} \\right)}} \\Gamma{\\left(\\frac{3}{4} \\right)}$$"
      ],
      "text/plain": [
       "                ⎛√2⋅x⎞       \n",
       "3⋅√2⋅√π⋅fresnels⎜────⎟⋅Γ(3/4)\n",
       "                ⎝ √π ⎠       \n",
       "─────────────────────────────\n",
       "           8⋅Γ(7/4)          "
      ]
     },
     "execution_count": 149,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integ.doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 155,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHMAAAAnCAYAAAA8XHcHAAAABHNCSVQICAgIfAhkiAAABURJREFU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\n",
      "text/latex": [
       "$$\\int_{0}^{\\infty} x^{y} e^{- x}\\, dx$$"
      ],
      "text/plain": [
       "∞          \n",
       "⌠          \n",
       "⎮  y  -x   \n",
       "⎮ x ⋅ℯ   dx\n",
       "⌡          \n",
       "0          "
      ]
     },
     "execution_count": 155,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integ = Integral(x**y*exp(-x), (x,0,oo))\n",
    "integ"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 156,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\begin{cases} \\Gamma{\\left(y + 1 \\right)} & \\text{for}\\: - \\Re{\\left(y\\right)} < 1 \\\\\\int_{0}^{\\infty} x^{y} e^{- x}\\, dx & \\text{otherwise} \\end{cases}$$"
      ],
      "text/plain": [
       "⎧ Γ(y + 1)    for -re(y) < 1\n",
       "⎪                           \n",
       "⎪∞                          \n",
       "⎪⌠                          \n",
       "⎨⎮  y  -x                   \n",
       "⎪⎮ x ⋅ℯ   dx    otherwise   \n",
       "⎪⌡                          \n",
       "⎪0                          \n",
       "⎩                           "
      ]
     },
     "execution_count": 156,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integ.doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 157,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Help on function doit in module sympy.integrals.integrals:\n",
      "\n",
      "doit(self, **hints)\n",
      "    Perform the integration using any hints given.\n",
      "    \n",
      "    Examples\n",
      "    ========\n",
      "    \n",
      "    >>> from sympy import Integral\n",
      "    >>> from sympy.abc import x, i\n",
      "    >>> Integral(x**i, (i, 1, 3)).doit()\n",
      "    Piecewise((2, Eq(log(x), 0)), (x**3/log(x) - x/log(x), True))\n",
      "    \n",
      "    See Also\n",
      "    ========\n",
      "    \n",
      "    sympy.integrals.trigonometry.trigintegrate\n",
      "    sympy.integrals.risch.heurisch\n",
      "    sympy.integrals.rationaltools.ratint\n",
      "    as_sum : Approximate the integral using a sum\n",
      "\n"
     ]
    }
   ],
   "source": [
    "help(Integral.doit)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Coming up Limits... "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\lim_{a \\rightarrow b}{f(x)}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 161,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA0AAAASCAYAAACAa1QyAAAABHNCSVQICAgIfAhkiAAAAHZJREFU\nKJFjYKACCGFgYJjMwMBwmIGB4RMDA8N/BgaGJYQ0XYAq/MzAwHCdWE2ODAwMqgwMDIwMDAwOuDSx\noPH3EzKVgYGBgYkYRaOaBlwTeuQGQDEDAwODBJS2ZGBgWABlv2FgYChBN6SBAZJ0cOEH5LiMzgAA\n6XoX52TB9a4AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1$$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 161,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(sin(x)/x, x, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 163,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACsAAAASCAYAAADCKCelAAAABHNCSVQICAgIfAhkiAAAAYpJREFU\nSInt1b1rFEEcxvEPEk06UeyEkMSztTcI2lkJIQSCEDT/QMDGwkrBQkgdRawsLBSMZYpA0iRRCfkD\nLEK8gKXvguf7pthJcPfmbvfONRCSB4ZlZn7zfGefnd1lDysJbRN9LWrqoaZnt1mHWhT149o/bqas\numYleI93+IgTkZq66pLtiBVL9itu4yhulgRPYhYbaOAzVjBRsK4b1o4SvMFhrOMHTudq6pqTbWAN\nD3EHD4JPEjZTJavJAMZC/1kJg1MRryNYwE+crJAVNYDnYexcWYOcRkPtlf/ByhucDWMvCwz6cRev\npOcwybUbVbCK0nmBp9LHNI4nkZohrOIYljCPT/iNAVxFbwGnLCuj/N1CTXr4N6TnsC6b7EzoT0b8\nLoe5W1WwWv0U/tY67mEQU5H5WrjORubOl/DvhJVR7G7hOD5IP+JvZZO9H/qXcmsu4pfOki1ilTKA\n67IvzbbBGXzHNzzCNObwB4+72Gw7VmmDXrxuYTCMRWkiX7CMEVzocrPtWAfa39oCSCeuK/kxTtUA\nAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\mathrm{NaN}$$"
      ],
      "text/plain": [
       "nan"
      ]
     },
     "execution_count": 163,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = x**2/exp(x)\n",
    "expr.subs(x,oo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}